← Do we need 4 different operations? Can two be enough?

• What mathematics can & can’t we do if we only use addition and

subtraction in a number system? • Do multiplication and division provide access to any additional mathematics?

← The “2” Camp

• Two may be enough (addition or subtraction) & (multiplication or

division).

• Only addition (multiplication is addition)

• If we’re tricky about it, we can treat some operations like others.

← The “4” Camp

• Hard to manage with less than 4.

• ½ * ¼ seems difficult (if not impossible) with only addition

• Might not be able to represent all numbers if we don’t have all the operations

← Further mathematics might be difficult/impossible

← If we start with 0-9 and +/-, then we can get negatives, we can get allthe integers, Natural, Integers. We’re missing rational numbers, irrationals,imaginary numbers.

← If we start with 0-9, and +/-/*/÷, we can get natural numbers, integers,rational numbers & irrational numbers (real numbers). imaginary?

← Do you think that you can use the standard algorithm for addition ormultiplication we are using today in other number systems. Why or why not?If yes, how?

• Can do

o Egyptian could do addition if you go horizontally and may

need regrouping.

o Greek could do addition

o Roman could do addition if you also kept track of subtractions.

• Can’t do o Systems without a zero placeholder, we can’t perform the standard algorithm for multiplication

← Week 3: Zeros and Fractions

← How has zero evolved in mathematics and as a number?

• Why was there a need for zero?

o Before there was a symbol for zero, there was just an empty space. This led to confusion, so a dot was used (sometimes the same as the punctuation for the end of a sentence). Zero began as being a symbol for a placeholder (first recognized as the absence of a quantity). Later recognized as a quantity. Our symbol for zero evolved from a tiny circle used as a place holder.

• How was zero named in different cultures?

o Set properties/identities of zero including the additive

property (any number plus zero is the original number) and the multiplicative identity (any number times zero is zero).

o Confusion existed when dividing by zero? (Is the answer zero,

one, something else?)

• What new mathematics became possible with each of these

changes in conceptions of zero?

o Algebra was developed (setting an equation equal to zero).

Originally, variables were used on both sides of an equation. Now all variables could be put on one side of an equation. Quadratic equations could be used to find roots of a quadratic equation (which is where the graph crosses the x-axis, if there are two real roots/one real root touches the axis/no real roots does not cross the axis). These are sometimes called the zeros of the equation.

← How have fractions evolved in mathematics?

• Why was there a need for fractions?

o Fractions began as representations of parts of a whole. There was a need for greater precision in measurement (feet -> half feet -> quarter feet, etc.). Sometimes fractions were given unique names instead of parts of a larger whole (cups, pints, quarts, gallons, etc.)

• How were fractions described and used in different cultures?

o Chinese avoided using unit fractions (7/3 was instead written

as 2 and 1/3).

• What is the difference between our current use of fractions and the unit fraction approach?

o Unit fractions consisted of a 1 in the numerator. Some

fractions were represented as the sum of unit fractions. Chinese were perhaps the first to go from unit fractions to more conventional notations of fractions (multiples of a small unit -> instead of ½ +1/4 you can now say ¾).

o Fractions could be written as decimals. This allowed easier

computations (which led to better understandings of square roots and pi). Percents (Per=per, cent=100) came from fractions. Base 60 used in time and navigation (from Babylonians).

← What are some of the logical difficulties that arise when you attempt todefine 0/0 to be 1 or 0?

← Issue – what does this problem represent?

• 0/0=0 is the same as 0 times what equals zero (of which there are an infinite number of answers). Zero times what equals 1 is not possible. Quotient Remainder Theorem

←Imagine a teacher shows her students a shortcut for dividing single digits by9. She says that all you have to do is to write the numerator as a repeatingdecimal (for example, 1/9=.1 repeating, that is 0.1111111..., and 4/9=.4repeating, that is 0.4444444...). A student raises their hand and asks if thatmeans that 9/9=.9 repeating. A different students says that this is impossiblebecause 9/9 has to be equal 1. What would say to these two students?

3/9=1/3=.333333…

0.333….+0.333…+0.333=0.999…

1/3+1/3+1/3=3/3=1 These are the same.

X=.99999…

10X=9.9999….

subtract these two things…

10X-X=9.9999…-0.99999…

9X=9

x=1

← Week 4: Negative and Imaginary Numbers

← Discussion on Negatives

← What shifts in people’s conceptions of numbers were required to

accept negative numbers?

• Previously, had counted number (thought of numbers) as objects

and measurements. There needed to be a purpose for negative numbers (reasons they had to exist). Had a new way of referring to negatives (as debts). People had to conceptualize something less than nothing. • There was confusion about where to put the negatives as compared to their positive counterparts. People had seen that negatives were coming up as solutions to equations, but previously they had ignored them (false root/fictitious solutions).

← One descriptions of negative numbers are numbers that are “less thannothing”. How would you explain or verify that negative numbers are “lessthan nothing”?

• A problem may exist with using 0 and nothing interchangeably.

• What other descriptions for negative numbers can you think of?

← _________________ 0

← | |

← | |

← | |

← \______________/

• Absolute value -> Distance from zero.

• Real world examples

o Owing money.

o Hot air balloon with negative numbers as sandbags.

← Have the number line displayed with both positive and negatives.

• Operations with negative numbers have defined rules (negative plus

negative is negative, negative times negative is positive, etc.). How can you justify those rules?

o Proof of Negative times Negative?

• Negative times Positive = Negative -> Negative *Negative is the

opposite of Negative times Positive, which is Positive.

o Bad things happen to good people is a bad thing, but a bad

thing happening to bad people is a good thing. Good things happening to good people is good. Good things happening to bad people is bad. (Friends of friends/friends of enemies…)o Logic based – Not going to the store -> not going, but not not going to the store is going.

o Using the two negative bars to form a plus sign.

 -n * -m = -(-n * m) = -(-nm)= nm← Discussion on Imaginary Numbers

• What shifts in people’s conceptions of numbers were required to

accept imaginary numbers?

o People had to understand that imaginary numbers were NOT

useless. These were not impossible solutions. They had to create the conception of imaginary numbers. This didn’t exist before and actually had to create the conceptions. Had to have an understanding of negative numbers, as well as an understanding of square roots.

← Of the different justifications of the Pythagorean Theorem in the text,

Had seen proof before. Use non-complex skills to arrive at formula. Didn’t have to manipulate picture.

← Week 6: Pi & Greek Geometry

← What is the mathematical definition/description of pi?

• Π=Ratio of circumference of a circle to its diameter

← How have people represented pi throughout history?

• Fractions or Mixed numbers - between 3+10/71 and 3+10/70, using

the symbolic symbol pi, decimal approximations, 22/7.

← Why has there been so attention paid to pi throughout history?

• Since pi is irrational, we’ll never know all the digits of the decimal. There also is no patterns found in the digits. Challenge to find more digits. Ratio is used so often (so popular) because of its relation to a circle. Also, possible discovers await if we can discover about the nature of irrational numbers within the digits.

← Week 7: Platonic Solids

← Coordinate Geometry• What is the mathematical definition/description of analytic geometry?

o Representation of shapes by equations.

o This provided a “numerical address” to shapes. Combined

algebra and geometry, links between the two.

o Related to the coordinate plane, need a concept of distance

and how to measure distance on a plane. Plots and trajectories.

• How has a coordinate system been used throughout history?

o Originally used to divide land into districts.

o Greece, 350 B.C., Apollonius plotted points that were a fixed

distance away from a given point to form a circle.

o A grid has been used to make maps and survey land.

o Pierre de Fermat in 1630, plot relationships between unknown

points.

• What new mathematics became available because of the use of

analytic geometry (and the perpendicular y-axis)?

o Helps with negative numbers (provided a boundary between

positive and negatives).

o Can represent more than one variable at a time now. Can use this to find areas and lengths.

o Can represent functions, this also led to discoveries in

Calculus. o

← Platonic Solids

• What are the names/properties of the Platonic Solids?

o Tetrahedron (4 sides)

o Dodecahedron (12 sides)

o Icosahedron (20 sides)

o Hexahedron (Cube) (6 sides)

o Octahedron (8 sides)

o All the faces are the same regular shapes. Every vertex has the same number of faces meeting (number of edges as well).

o The angles at each vertex must add to less than 360 degrees (or else it’ll be flat.)

o Symmetry around each vertex.

o At least three faces must meet at each vertex.

• What associations did the Pythagoreans make with these solids?

o 4 elements

 Tetrahedron (4) -> Fire (looks like a triangle/four

necessities for fire)

 Hexahedron (6) -> Earth (belief earth was flat/set on a

surface, won’t roll/six major land forms on Earth)

 Octahedron (8) -> Air

 Icosahedrons (20) -> Water (most spherical, most likely

to roll out of your hand)

 Dodecahedron (12) -> Universe

• What is the distinction between Platonic solids and Archimedean

Solids? o Archimedean solids have the same properties as Platonic solids, but the faces don’t all have to be the same one shape. Each type of shape has to be congruent to all of the same shapes in the figure.

← Week 8: Non-Euclidean Geometries

← Non-Euclidean Geometry

• How is non-Euclidean Geometry different than Euclidean Geometry?

Give an example.

o Euclidean geometry is on the surface of a plane (non-

Euclidean is not). An example was Riemann which was on the surface of a sphere.

o A sphere has no parallel lines, so the parallel postulate

doesn’t hold. Triangle angle measures add to over 180 degrees. Pi is different

• Which of Euclid’s 5 postulates is not true in spherical geometry.

Explain.

o Postulate 2 doesn’t. Lines are finite (can be measured).

o Postulate 5 doesn’t. There are no parallel lines.

• Explain why the statement, “Euclidean Geometry is better than non-

Euclidean geometries.” is an unfair statement to make.

o While we may know Euclidean and be more familiar with it,

other subject areas rely on and use different kinds of geometries. (For example, spherical geometry is good for astronomy.)

← Projective Geometry

• What motivated the development of projective geometry?

o Based on a need from philosophers, artists, and scientists. Wanted to represent three dimensional scenes on paper (in two dimensions). Wanted to look at the relationship between distance on the paper versus distance in real life.

• What is the principle of duality? Explain with an example.

o Point-Line duality - For a statement that includes the words

“line” and “point”, interchanging the words still produces a true statement. “Points are collinear if they all lie on the same line” and “Lines are concurrent if they all intersect at the same point.”

← Perspective in Art

• How did the use of perspective change the perception and purpose of art in society.

o Non-perspective art (such as Egyptian) was used to tell a

story (or portray a message) for record keeping. Perspective allowed art to be more enjoyable and more abstract.

• Perspective drawing allowed better blueprints for planning.

o In art, perspective fed a realism movement. Going from

cartoonish icons to more real world descriptions. Perspective allowed showing depth in space.

← Week 10: Linear Equations

• What is proof?

o Proof (provides/is) evidence that something has to be true (or

not true). Provides a reason that something is true (or not true).

o I proved the Pythagorean Theorem means…

 This means I took a given, and used supporting facts in

a step by step (sequential) manner which can be followed to supports a conclusion, and arrived at a conclusion. • How is proof used in mathematics? In math education? In real life? How are these the same and how are they different?

o Mathematics – tend to think it’s something new, uses

theorems and theories

o Math Education – tend to think it’s something already done,

use manipulatives (not just words), proof can also mean convincing someone, but not necessarily a formal proof. Visual signals can be convincing. Understanding a proof can lead to greater understanding of what was proved.

o Real life – the scope (one instance vs. all instances), trying to get evidence to be convinced. Visual signals can be convincing.

o All of these areas tend to be about convincing someone that

something is true.

• Why do we use proof in these different contexts?

• What is the difference between a deductive proof and an inductive

proof?

← Discussion of Reading (Writing Algebra)

• What is your own definition/description of algebra? Is there a formal

definition of algebra? (If so, what?)

o Algebra (colloquial) – equations and expressions that include

symbols and variables (that represent quantities). Can use algebra to manipulate equations to isolate a variable. Solving for an unknown. Manipulation of polynomials that can also represent graphs.

o Algebra (formal) A part of mathematics in which letters and

other general symbols, are used to represent numbers and quantities in formulae and equations. o Algebra problems (formal) – regardless of how it is written, it is a questions about numerical operations and relations in which an unknown quantity must be deduced from known ones.

• What are the characteristics of algebra that distinguish it from other

branches of mathematics?

o Equations are already set equal to something, as opposed to

being a variable.

o Algebra uses arithmetic to find an answer, but arithmetic

doesn’t use algebra to find an answer.

o Algebra is a generalization of arithmetic.

o Geometry involves shapes/pictures.

o Algebra may be used in geometry solutions.

• How is a symbolic style different from a rhetorical style in algebra?

Give an example of each, and state one advantage of each approach.

X^2+5x+6=0 <- Symbolic algebra

S + 5T + 6 is 0. <- Syncopated algebra

A square and five things and six is equal to zero. <- Rhetorical style uses words.

← Discussion of Reading (Solving Linear Equations)

• What is the false position method? Give some examples of the

method.

o Consider the problem, a quantity and its fifth becomes 24.

o A third of a quantity is taken from half of a quantity and

becomes 18.

• Solve these problems with our traditional algorithm.

o How is the mathematics used in these two solutions the same? Different? Are there any similarities between this method other methods we use today.

o What are the different approaches we use to solve these

problems today?

o Will the false position method work for all first degree algebraic problems? How do you know?

← Week 11: Polynomials and Patterns

← Sketch 10 – Quadratic Equations

• What is meant by “quadratic” equations?

o Dictionary.com – equations containing a single variable to the

second degree. (It’s general form is ax^2+bx+c=0, where a is not equal to zero.).

o Wikipedia – a polynomial of degree 2.

o Comes from the Latin word quadractum which means

“square”.

• What are different methods that have been used to solve quadratic equations in the past?

o Solving an equation – finding where the graph of the equation

passes through the x-axis. This is also finding the values of x, such that when you plug them back into the equation, the equation comes out to zero.

o Different approaches – graphic, algebraic (factoring, quadratic

formula, completing the square, difference of squares), geometric

• What are some of the quadratic equations and solutions (current

method & al-Khwarizmi’s method) you came up with?

X^2 + 8X = 20 x 8 •

x •

• x 4 • 8 • x •

• 4

X^2+8X+16=20+16

(X+4)^2=36

X+4=6

X=2

← Week 12: Calculus

← Imagine you are trying to find different ways of calculating the area ofthe shaded region of the parabola. What are different methods you couldcome up with?

• Integrals – taking the integral of the first point minus the integral of the second point.

• Area of the parabola, then subtracting the area underneath it. This would be using the rectangle approximation.

o Breaking up into multiple figure. (Would be approximating).

o Playdough – roll it out to a uniform thickness.

← Imagine you are trying to find different ways of determining the speedof a falling object. What different methods can you come up with?

• Use a speed gun (radar gun)

• Measure falling distance and falling time. (9.8m/square second)

• Compare impact markings to KNOWN impact marking.

• Vertex formula

← What is origin of the word “calculus”?

• Originally from the word “stone” or ”pebble”.

← What does calculus mean in mathematics?

• Rate of change.

← What is the distinction between calculus and other mathematical

subjects?

• Calculus takes into consideration change. Focuses on limits,

derivates, integrals, functions, and infinite series.

← What were the origins of calculus?

← What problems did calculus help solve?

• Helps us solve things that go on to infinity and don’t have an ending

point. Things that go to an infinitely small infinity. Also addresses rates of change and fast things are moving.